The exact order of normal approximation in bivariate renewal theory

Abstract

Equivalence of rates of convergence in the central limit theorem between the vector of maximum sums and the corresponding first-passage variables is established. The bivariate case is studied. Analogous results about the equivalence between the vector of partial sums and corresponding renewal variables are also given and as a consequence we obtain a generalization of a theorem of Hunter (1974). Extension of the main result to more general first-passage times is also developed.